1. ## disjoint intervals

Let F be a collection of disjoint intervals of real numbers. This means that if I and J are two distinct intervals in F, then the intersection of I and J is zero. Prove that the set F is countable.

2. Originally Posted by friday616
Let F be a collection of disjoint intervals of real numbers. This means that if I and J are two distinct intervals in F, then the intersection of I and J is empty?. Prove that the set F is countable.
Between any two real numbers there is a rational number.
Properly defined a closed interval looks like $\displaystyle \left[ {a,b} \right] = \left\{ {x:a < b\;\& \;a \leqslant x \leqslant b} \right\}$.
Thus in each closed interval in the collection $\displaystyle \mathcal{F}$ there is a rational number.
Because the set of rationals is countable the collection $\displaystyle \mathcal{F}$ is countable.

3. Originally Posted by friday616
Let F be a collection of disjoint intervals of real numbers. This means that if I and J are two distinct intervals in F, then the intersection of I and J is zero. Prove that the set F is countable.
Do you consider single point sets, such as $\displaystyle [1,1]$ to be intervals?

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# disjoint time intervals meaning

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